Opuscula Math. 40, no. 5 (2020), 617-627
https://doi.org/10.7494/OpMath.2020.40.5.617

Opuscula Mathematica

# On 2-rainbow domination number of functigraph and its complement

Ebrahim Vatandoost

Abstract. Let $$G$$ be a graph and $$f:V (G)\rightarrow P(\{1,2\})$$ be a function where for every vertex $$v\in V(G)$$, with $$f(v)=\emptyset$$ we have $$\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}$$. Then $$f$$ is a $$2$$-rainbow dominating function or a $$2RDF$$ of $$G$$. The weight of $$f$$ is $$\omega(f)=\sum_{v\in V(G)} |f(v)|$$. The minimum weight of all $$2$$-rainbow dominating functions is $$2$$-rainbow domination number of $$G$$, denoted by $$\gamma_{r2}(G)$$. Let $$G_1$$ and $$G_2$$ be two copies of a graph G with disjoint vertex sets $$V(G_1)$$ and $$V(G_2)$$, and let $$\sigma$$ be a function from $$V(G_1)$$ to $$V(G_2)$$. We define the functigraph $$C(G,\sigma)$$ to be the graph that has the vertex set $$V(C(G,\sigma)) = V(G_1)\cup V(G_2)$$, and the edge set $$E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}$$. In this paper, $$2$$-rainbow domination number of the functigraph of $$C(G,\sigma)$$ and its complement are investigated. We obtain a general bound for $$\gamma_{r2}(C(G,\sigma))$$ and we show that this bound is sharp.

Keywords: 2-rainbow domination number, functigraph, complement, cubic graph.

Mathematics Subject Classification: 05C69, 05C75.

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• Imam Khomeini International University, Department of Basic Science, Nouroozian Street, Qazvin, Iran
• Ebrahim Vatandoost (corresponding author)
• Imam Khomeini International University, Department of Basic Science, Nouroozian Street, Qazvin, Iran
• Communicated by Dalibor Fronček.
• Revised: 2020-06-04.
• Accepted: 2020-07-07.
• Published online: 2020-10-10.